Gaussian Random Processes de I A Ibragimov

Gaussian Random Processes

I A Ibragimov

Y.A. Rozanov

A B Aries

The book deals mainly with three problems involving Gaussian stationary processes. The first problem consists of clarifying the conditions for mutual absolute continuity (equivalence) of probability distributions of a "random process segment" and of finding effective formulas for densities of the equiva lent distributions. Our second problem is to describe the classes of spectral measures corresponding in some sense to regular stationary processes (in par ticular, satisfying the well-known "strong mixing condition") as well as to describe the subclasses associated with "mixing rate". The third problem involves estimation of an unknown mean value of a random process, this random process being stationary except for its mean, i. e. , it is the problem of "distinguishing a signal from stationary noise". Furthermore, we give here auxiliary information (on distributions in Hilbert spaces, properties of sam ple functions, theorems on functions of a complex variable, etc. ). Since 1958 many mathematicians have studied the problem of equivalence of various infinite-dimensional Gaussian distributions (detailed and sys tematic presentation of the basic results can be found, for instance, in [23]). In this book we have considered Gaussian stationary processes and arrived, we believe, at rather definite solutions. The second problem mentioned above is closely related with problems involving ergodic theory of Gaussian dynamic systems as well as prediction theory of stationary processes.

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Cuprins

I Preliminaries.- I.1 Gaussian Probability Distribution in a Euclidean Space.- I.2 Gaussian Random Functions with Prescribed Probability Measure.- I.3 Lemmas on the Convergence of Gaussian Variables.- I.4 Gaussian Variables in a Hilbert Space.- I.5 Conditional Probability Distributions and Conditional Expectations.- I.6 Gaussian Stationary Processes and the Spectral Representation.- II The Structures of the Spaces H(T) and LT(F).- II. 1 Preliminaries.- II.2 The Spaces L+(F) and L-(F).- II.3 The Construction of Spaces LT(F) When T Is a Finite Interval.- II.4 The Projection of L+(F) on L-(F).- II.5 The Structure of the ?-algebra of Events U(T).- III Equivalent Gaussian Distributions and their Densities.- III.1 Preliminaries.- III.2 Some Conditions for Gaussian Measures to be Equivalent.- III.3 General Conditions for Equivalence and Formulas for Density of Equivalent Distributions.- III.4 Further Investigation of Equivalence Conditions.- IV Conditions for Regularity of Stationary Random Processes.- IV.1 Preliminaries.- IV.2 Regularity Conditions and Operators Bt.- IV.3 Conditions for Information Regularity.- IV.4 Conditions for Absolute Regularity and Processes with Discrete Time.- IV.5 Conditions for Absolute Regularity and Processes with Continuous Time.- V Complete Regularity and Processes with Discrete Time.- V.l Definitions and Preliminary Constructions with Examples.- V.2 The First Method of Study: Helson—Sarason’s Theorem.- V.3 The Second Method of Study: Local Conditions.- V.4 Local Conditions (continued).- V.5 Corollaries to the Basic Theorems with Examples.- V.6 Intensive Mixing.- VI Complete Regularity and Processes with Continuous Time.- VI.1 Introduction.- VI.2 The Investigation of a Particular Function ?(T;µ).- VI.3 The Proof of the Basic Theorem onNecessity.- VI.4 The Behavior of the Spectral Density on the Entire Line.- VI.5 Sufficiency.- VI.6 A Special Class of Stationary Processes.- VII Filtering and Estimation of the Mean.- VII.1 Unbiased Estimates.- VII.2 Estimation of the Mean Value and the Method of Least Squares.- VII.3 Consistent Pseudo-Best Estimates.- VII.4 Estimation of Regression Coefficients.- References.